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Properties

Label 2-21e2-63.13-c2-0-6
Degree $2$
Conductor $441$
Sign $0.963 - 0.266i$
Analytic cond. $12.0163$
Root an. cond. $3.46646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.840 − 1.45i)2-s + (−2.99 − 0.143i)3-s + (0.585 − 1.01i)4-s + (−2.03 − 1.17i)5-s + (2.31 + 4.48i)6-s − 8.69·8-s + (8.95 + 0.857i)9-s + 3.94i·10-s + (3.10 + 5.37i)11-s + (−1.89 + 2.95i)12-s + (−21.3 − 12.3i)13-s + (5.91 + 3.80i)15-s + (4.97 + 8.61i)16-s + 22.5i·17-s + (−6.28 − 13.7i)18-s + 10.7i·19-s + ⋯
L(s)  = 1  + (−0.420 − 0.728i)2-s + (−0.998 − 0.0476i)3-s + (0.146 − 0.253i)4-s + (−0.406 − 0.234i)5-s + (0.385 + 0.747i)6-s − 1.08·8-s + (0.995 + 0.0952i)9-s + 0.394i·10-s + (0.282 + 0.488i)11-s + (−0.158 + 0.246i)12-s + (−1.64 − 0.948i)13-s + (0.394 + 0.253i)15-s + (0.310 + 0.538i)16-s + 1.32i·17-s + (−0.349 − 0.765i)18-s + 0.568i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.963 - 0.266i$
Analytic conductor: \(12.0163\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1),\ 0.963 - 0.266i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.472267 + 0.0639696i\)
\(L(\frac12)\) \(\approx\) \(0.472267 + 0.0639696i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 + 0.143i)T \)
7 \( 1 \)
good2 \( 1 + (0.840 + 1.45i)T + (-2 + 3.46i)T^{2} \)
5 \( 1 + (2.03 + 1.17i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.10 - 5.37i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (21.3 + 12.3i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 22.5iT - 289T^{2} \)
19 \( 1 - 10.7iT - 361T^{2} \)
23 \( 1 + (-4.27 + 7.39i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.1 - 27.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.44 - 0.833i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 39.5T + 1.36e3T^{2} \)
41 \( 1 + (-27.9 - 16.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (2.15 + 3.73i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-42.0 + 24.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.09T + 2.80e3T^{2} \)
59 \( 1 + (-91.7 - 52.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (20.0 - 11.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.29 + 2.24i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 66.6T + 5.04e3T^{2} \)
73 \( 1 - 21.3iT - 5.32e3T^{2} \)
79 \( 1 + (-51.5 - 89.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-10.0 + 5.80i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 13.8iT - 7.92e3T^{2} \)
97 \( 1 + (-13.0 + 7.54i)T + (4.70e3 - 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.77450574687561731505490087344, −10.26432247636233561932265409891, −9.590909794628870188890471411916, −8.267532005708166576187172034235, −7.20964079717448011243442998571, −6.15955377010018166635189809801, −5.24383988387844193190403849618, −4.10330961451645763408540574323, −2.42265297661535163856279819949, −1.01878666613179560610191376585, 0.31788023539590377607805553420, 2.64696463282742982448995492943, 4.20896947836471693945547939623, 5.30125762908486096007738484148, 6.43388771727148249271355810710, 7.18861060501634142611193622680, 7.70492588843661008155921359967, 9.199234134248255885330751624502, 9.707028626394180752645437876627, 11.15358301890118977741088660974

Graph of the $Z$-function along the critical line